Multiphase Coriolis Flowmeter

ABSTRACT

A flowmeter is disclosed. The flowmeter includes a vibratable flowtube, and a driver connected to the flowtube that is operable to impart motion to the flowtube. A sensor is connected to the flowtube and is operable to sense the motion of the flowtube and generate a sensor signal. A controller is connected to receive the sensor signal. The controller is operable to determine a first flow rate of a first phase within a two-phase flow through the flowtube and determine a second flow rate of a second phase within the two-phase flow.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.11/344,140, filed Feb. 1, 2006 and titled “MULTIPHASE CORIOLISFLOWMETER,” which is a continuation of U.S. application Ser. No.10/737,459, filed Feb. 9, 2004 and titled “MULTIPHASE CORIOLISFLOWMETER,” which claims priority under 35 USC §119(e) to U.S. PatentApplication Ser. No. 60/445,795, filed on Feb. 10, 2003, and titledMULTIPHASE CORIOLIS FLOWMETER. This application also claims priority toU.S. Application Ser. No. 60/452,934, filed on Mar. 10, 2003, and titledMULTIPHASE CORIOLIS FLOWMETER. These applications are herebyincorporated by reference.

TECHNICAL FIELD

This description relates to flowmeters.

BACKGROUND

Flowmeters provide information about materials being transferred througha conduit, or flowtube. For example, mass flowmeters provide anindication of the mass of material being transferred through a conduit.Similarly, density flowmeters, or densitometers, provide an indicationof the density of material flowing through a conduit. Mass flowmetersalso may provide an indication of the density of the material, andtherefore an indication of the volumetric flow rate.

For example, Coriolis-type mass flowmeters are based on the Corioliseffect, in which material flowing through a conduit becomes aradially-travelling mass that is affected by a Coriolis force andtherefore experiences an acceleration. Many Coriolis-type massflowmeters induce a Coriolis force by sinusoidally oscillating a conduitabout a pivot axis orthogonal to the length of the conduit. In such massflowmeters, the Coriolis reaction force experienced by the travelingfluid mass is transferred to the conduit itself and is manifested as adeflection or offset of the conduit in the direction of the Coriolisforce vector in the plane of rotation.

SUMMARY

According to one general aspect, a flowmeter includes a vibratableflowtube, a driver connected to the flowtube and operable to impartmotion to the flowtube, a sensor connected to the flowtube and operableto sense the motion of the flowtube and generate a sensor signal, and acontroller connected to receive the sensor signal, the controller beingoperable to determine a first flow rate of a first phase within atwo-phase flow through the flowtube and determine a second flow rate ofa second phase within the two-phase flow.

Implementations may include one or more of the following features. Forexample, the first phase may include a gas and the second phase mayinclude a liquid.

The controller may be operable to input an apparent density of thetwo-phase flow detected by the flowmeter and output a corrected densityof the two-phase flow. The controller may be operable to correct theapparent density based on a theoretical relationship between theapparent density and the corrected density, or based on an empiricalrelationship between the apparent density and the corrected density(such as, for example, a table storing relationships between theapparent density and the corrected density).

The controller may be operable to input an apparent mass flow rate ofthe two-phase flow detected by the flowmeter and output a corrected massflow rate of the two-phase flow. The controller may be operable tocorrect the apparent mass flow rate based on a theoretical or empiricalrelationship, such as a tabular relationship, between the apparent massflow rate and the corrected mass flow rate.

The controller may be operable to input an apparent first phase fractionof the two-phase flow detected by the flowmeter that defines an amountof the first phase in the two-phase flow and output a corrected firstphase fraction of the two-phase flow. The controller may be operable toinput a first phase fraction of the two-phase flow detected by a phasefraction sensor that is external to the flowmeter.

The controller may be operable to determine the first flow rate and thesecond flow rate based on corrected values for a detected density anddetected mass flow rate of the two-phase flow. The controller may beoperable to determine the first flow rate and the second flow rate basedon a corrected value for a detected first phase fraction that defines anamount of the first phase in the two-phase flow. The controller may beoperable to determine the first flow rate and the second flow rate basedon densities of the first phase and the second phase, respectively.

The controller may be operable to determine a first superficial velocityof the first phase and a second superficial velocity of the secondphase, based on the first flow rate and the second flow rate,respectively. The controller may be operable to determine a flow regimeof the two-phase flow, based on the first superficial velocity and thesecond superficial velocity. The controller may be operable to determinea slip velocity between the first phase and the second phase, based onan average velocity of the first phase and an average velocity of thesecond phase. The controller may be operable to provide corrections tothe first flow rate and the second flow rate, based on the first andsecond superficial velocities, the determined flow regime, or the slipvelocity, to thereby obtain a corrected first flow rate and a correctedsecond flow rate.

According to another general aspect, a method includes determining abulk density of a two-phase flow through a flowtube, the two-phase flowincluding a first phase and a second phase, determining a bulk mass flowrate of the two-phase flow, and determining a first mass flow rate ofthe first phase, based on the bulk density and the bulk mass flow rate.

Implementations may include one or more of the following features. Forexample, a second mass flow rate of the second phase may be determined,based on the bulk density and the bulk mass flow rate. In determiningthe bulk density, an apparent bulk density of the two-phase flow may bedetermined, and the apparent bulk density may be corrected to obtain thebulk density.

In correcting the apparent bulk density, the apparent bulk density maybe input into a theoretical relationship that relates the apparent bulkdensity to a corrected bulk density, or may be input into an empiricalrelationship that relates the apparent bulk density to a corrected bulkdensity.

In correcting the apparent bulk density, a first density of the firstphase may be input. A first phase fraction of the two-phase flow may bedetermined, based on the bulk density, the first density of the firstphase, and a second density of the second phase. In determining thefirst mass flow rate of the first phase, the first mass flow rate may bedetermined based on the first phase fraction and the first density.

A first superficial velocity of the first phase and a second superficialvelocity of the second phase may be determined, based on the first massflow rate and the second mass flow rate, respectively. A flow regime ofthe two-phase flow may be determined, based on the first superficialvelocity and the second superficial velocity. A slip velocity betweenthe first phase and the second phase may be determined, based on anaverage velocity of the first phase and an average velocity of thesecond phase. Corrections may be provided to the first flow rate and thesecond flow rate, based on the first and second superficial velocities,the determined flow regime, or the slip velocity.

The first phase may include a gas and the second phase may include aliquid.

According to another general aspect, a flowmeter controller includes adensity correction system operable to input an apparent density of atwo-phase flow and output a corrected density of the two-phase flow, thetwo-phase flow including a first phase and a second phase, a mass flowrate correction system operable to input an apparent mass flow rate ofthe two-phase flow and output a corrected mass flow rate of thetwo-phase flow, and a flow component mass flow rate determination systemoperable to determine a first mass flow rate of the first phase, basedon the corrected density and the corrected mass flow rate.

Implementations may include one or more of the following features. Forexample, the flow component mass flow rate determination system may beoperable to determine a second mass flow rate of the second phase, basedon the corrected density and the corrected mass flow.

The first phase may include a liquid and the second phase may include agas. A phase fraction determination system may be included that isoperable to determine a corrected phase fraction of the two-phase flow,wherein the flow component mass flow rate determination system may beoperable to determine the first flow rate and the second flow rate basedon the corrected phase fraction. The phase fraction determination systemmay be a void fraction determination system that determines an amount ofthe gas in the two-phase flow.

A superficial velocity determination system may be included that isoperable to determine a first superficial velocity of the first phaseand a second superficial velocity of the second phase. The flowmetercontroller may include a flow regime determination system operable todetermine a flow regime of the two-phase flow.

The flow regime determination system may be further operable todetermine a phase slip velocity with respect to an average velocity ofthe first phase and an average velocity of the second phase. The flowcomponent mass flow rate determination system may be operable to improvethe determination of the first mass flow rate and the second mass flowrate, based on the first and second superficial velocities, the flowregime, or the phase slip velocity.

The details of one or more implementations are set forth in theaccompanying drawings and the description below. Other features will beapparent from the description and drawings, and from the claims.

DESCRIPTION OF DRAWINGS

FIG. 1A is an illustration of a Coriolis flowmeter using a bentflowtube.

FIG. 1B is an illustration of a Coriolis flowmeter using a straightflowtube.

FIG. 2 is a block diagram of a Coriolis flowmeter.

FIG. 3 is a flowchart illustrating an operation of the coriolisflowmeter of FIG. 2.

FIG. 4 is a flowchart illustrating techniques for determining liquid andgas flow rates for a two-phase flow.

FIGS. 5A and 5B are graphs illustrating a percent error in a measurementof void fraction and liquid fraction, respectively.

FIG. 6 is a graph illustrating a mass flow error as a function of a dropin density for a flowtube having a particular orientation and over aselected flow range.

FIG. 7 is a flowchart illustrating techniques for correcting densitymeasurements.

FIG. 8 is a table showing a relationship between an apparent densitydrop and an apparent mass flow rate of the two-phase flow.

FIG. 9 is a flowchart illustrating techniques for determining voidfraction measurements.

FIG. 10 is a flowchart illustrating techniques for determining correctedmass flow rate measurements.

FIG. 11 is a table showing a relationship between an apparent mass flowrate and a corrected density drop of the two-phase flow.

FIGS. 12-14 are graphs illustrating examples of density corrections fora number of flowtubes.

FIGS. 15-20 are graphs illustrating examples of mass flow ratecorrections for a number of flowtubes.

DETAILED DESCRIPTION

Types of flowmeters include digital flowmeters. For example, U.S. Pat.No. 6,311,136, which is hereby incorporated by reference, discloses theuse of a digital flowmeter and related technology including signalprocessing and measurement techniques. Such digital flowmeters may bevery precise in their measurements, with little or negligible noise, andmay be capable of enabling a wide range of positive and negative gainsat the driver circuitry for driving the conduit. Such digital flowmetersare thus advantageous in a variety of settings. For example,commonly-assigned U.S. Pat. No. 6,505,519, which is incorporated byreference, discloses the use of a wide gain range, and/or the use ofnegative gain, to prevent stalling and to more accurately exercisecontrol of the flowtube, even during difficult conditions such astwo-phase flow (e.g., a flow containing a mixture of liquid and gas).

Although digital flowmeters are specifically discussed below withrespect to, for example, FIGS. 1 and 2, it should be understood thatanalog flowmeters also exist. Although such analog flowmeters may beprone to typical shortcomings of analog circuitry, e.g., low precisionand high noise measurements relative to digital flowmeters, they alsomay be compatible with the various techniques and implementationsdiscussed herein. Thus, in the following discussion, the term“flowmeter” or “meter” is used to refer to any type of device and/orsystem in which a Coriolis flowmeter system uses various control systemsand related elements to measure a mass flow, density, and/or otherparameters of a material(s) moving through a flowtube or other conduit.

FIG. 1A is an illustration of a digital flowmeter using a bent flowtube102. Specifically, the bent flowtube 102 may be used to measure one ormore physical characteristics of, for example, a (traveling) fluid, asreferred to above. In FIG. 1A, a digital transmitter 104 exchangessensor and drive signals with the bent flowtube 102, so as to both sensean oscillation of the bent flowtube 102, and to drive the oscillation ofthe bent flowtube 102 accordingly. By quickly and accurately determiningthe sensor and drive signals, the digital transmitter 104, as referredto above, provides for fast and accurate operation of the bent flowtube102. Examples of the digital transmitter 104 being used with a bentflowtube are provided in, for example, commonly-assigned U.S. Pat. No.6,311,136.

FIG. 1B is an illustration of a digital flowmeter using a straightflowtube 106. More specifically, in FIG. 1B, the straight flowtube 106interacts with the digital transmitter 104. Such a straight flowtubeoperates similarly to the bent flowtube 102 on a conceptual level, andhas various advantages/disadvantages relative to the bent flowtube 102.For example, the straight flowtube 106 may be easier to (completely)fill and empty than the bent flowtube 102, simply due to the geometry ofits construction. In operation, the bent flowtube 102 may operate at afrequency of, for example, 50-110 Hz, while the straight flowtube 106may operate at a frequency of, for example, 300-1,000 Hz. The bentflowtube 102 represents flowtubes having a variety of diameters, and maybe operated in multiple orientations, such as, for example, in avertical or horizontal orientation.

Referring to FIG. 2, a digital mass flowmeter 200 includes the digitaltransmitter 104, one or more motion sensors 205, one or more drivers210, a flowtube 215 (which also may be referred to as a conduit, andwhich may represent either the bent flowtube 102, the straight flowtube106, or some other type of flowtube), and a temperature sensor 220. Thedigital transmitter 104 may be implemented using one or more of, forexample, a processor, a Digital Signal Processor (DSP), afield-programmable gate array (FPGA), an ASIC, other programmable logicor gate arrays, or programmable logic with a processor core. It shouldbe understood that, as described in U.S. Pat. No. 6,311,136, associateddigital-to-analog converters may be included for operation of thedrivers 210, while analog-to-digital converters may be used to convertsensor signals from the sensors 205 for use by the digital transmitter104.

The digital transmitter 104 generates a measurement of, for example,density and/or mass flow of a material flowing through the flowtube 215,based at least on signals received from the motion sensors 205. Thedigital transmitter 104 also controls the drivers 210 to induce motionin the flowtube 215. This motion is sensed by the motion sensors 205.

Density measurements of the material flowing through the flowtube arerelated to, for example, the frequency of the motion of the flowtube 215that is induced in the flowtube 215 by a driving force supplied by thedrivers 210, and/or to the temperature of the flowtube 215. Similarly,mass flow through the flowtube 215 is related to the phase and frequencyof the motion of the flowtube 215, as well as to the temperature of theflowtube 215.

The temperature in the flowtube 215, which is measured using thetemperature sensor 220, affects certain properties of the flowtube, suchas its stiffness and dimensions. The digital transmitter 104 maycompensate for these temperature effects. Also in FIG. 2, a pressuresensor 225 is in communication with the transmitter 104, and isconnected to the flowtube 215 so as to be operable to sense a pressureof a material flowing through the flowtube 215.

It should be understood that both the pressure of the fluid entering theflowtube 215 and the pressure drop across relevant points on theflowtube may be indicators of certain flow conditions. Also, whileexternal temperature sensors may be used to measure the fluidtemperature, such sensors may be used in addition to an internalflowmeter sensor designed to measure a representative temperature forflowtube calibrations. Also, some flowtubes use multiple temperaturesensors for the purpose of correcting measurements for an effect ofdifferential temperature between the process fluid and the environment(e.g., a case temperature of a housing of the flowtube). As discussed inmore detail below, one potential use for the inlet fluid temperature andpressure measurements is to calculate the actual densities of a liquidand gas in a two-phase flow, based on predefined formulae.

A liquid fraction probe 230 refers to a device for measuring a volumefraction of liquid, e.g., water, when a liquid in the flowtube 215includes water and another fluid, such as oil. Of course, such a probe,or similar probes, may be used to measure the volume fraction of a fluidother than water, if such a measurement is preferred or if the liquiddoes not include water. In the below description, a measured liquid isgenerally assumed to be water for the purposes of example, so that theliquid fraction probe 230 is generally referred to as a water fractionprobe 230, or a water-cut probe 230.

A void fraction sensor 235 measures a percentage of a material in theflowtube 215 that is in gaseous form. For example, water flowing throughthe flowtube 215 may contain air, perhaps in the form of bubbles. Such acondition, in which the material flowing through the flowtube 215contains more than one material is generally referred to as “two-phaseflow.” In particular, the term “two-phase flow” may refer to a liquidand a gas; however, “two-phase flow” also may refer to othercombinations of materials, such as two liquids (e.g., oil and water).

Various techniques, represented generally in FIG. 2 by the void fractionsensor 235, exist for measuring the gas void fraction in a two-phaseflow of liquid and gas. For example, various sensors or probes existthat may be inserted into the flow to determine a gas void fraction. Asanother example, a venturi tube (i.e., a tube with a constricted throatthat determines fluid pressures and velocities by measurement ofdifferential pressures generated at the throat as a fluid traverses thetube), relying on the fact that gas generally moves with a highervelocity than liquid(s) through a restriction, may be used to determinea pressure gradient and thereby allow a determination of the gas voidfraction. Measurements of gas void fractions also may be obtained usingequipment that is wholly external to the flowtube. For example, sonarmeasurements may be taken to determine gas void fraction. As a specificexample of such a sonar-based system, the SONARtrac™ gas void fractionmonitoring system produced by CiDRA Corporation of Wallingford, Conn.may be used.

In this description, an amount of gas in a flowing fluid, measured bythe void fraction sensor or otherwise determined, is referred to as voidfraction or α, and is defined as α=volume of gas/total volume=volume ofgas/(volume of liquid+volume of gas). Accordingly, a quantity referredto herein as the liquid fraction is defined as 1−α.

In many applications where mass flow measurements are required, the voidfraction of the flow can be as high as 20, 30, 40% or more. However,even at very small void fractions of 0.5%, the fundamental theory behindthe coriolis flowmeter becomes less applicable.

Moreover, a presence of gas in the fluid flow also may affect ameasurement of a density of the fluid flow, generally causing thedensity measurement to read lower. That is, it should be understood thata density ρ_(liquid) of a liquid flowing by itself through a flowtubewill be higher than an actual density ρ_(true) of a two-phase flowcontaining the liquid and a gas, since a density of the gas (e.g., air)will generally be lower than a density of the liquid (e.g., water) inthe two-phase flow. In other words, there is a density reduction whengas is added to a liquid flow that previously contained only the liquid.

Beyond this physical phenomenon, a coriolis meter measuring a two-phasefluid flow containing gas may output a density reading ρ_(apparent) thatis an ostensible measurement of the bulk density of the two-phase flow(e.g., of the water and air combined). This raw measurement ρ_(apparent)will generally be different (lower) than the actual bulk densityρ_(true) of the two-phase flow. For example, the resonant frequency usedby the flowmeter may be artificially high, due to relative motion of thegas in the fluid flow, which would cause the density measurement to readlow. It should be understood that many conventional prior art flowmeterswere unconcerned with this problem, since most such coriolis meters failto continue operating (e.g. stall or output inaccurate measurements) ateven the slightest amounts of void fraction.

U.S. Pat. No. 6,505,519, which is incorporated by reference above,discloses that such a variation of ρ_(apparent) (i.e., an indicated rawor bulk density reading of a two-phase flow that is output by a coriolisflowmeter) from ρ_(true) (i.e., an actual raw or bulk density of thetwo-phase flow) may be characterized by a variety of techniques. As aresult, a measured ρ_(apparent) may be corrected to obtain an actualbulk density ρ_(corrected), which is, at least approximately, equal toρ_(true).

Somewhat similarly, an indicated raw or bulk mass flow rateMF_(apparent) (i.e., a mass flow rate of the entire two-phase flow)measured by a coriolis flowmeter may be different by a predictable orcharacterizable amount from an actual bulk mass flow rate MF_(true). Itshould be understood that correction techniques for corrected bulk massflow rate MF_(true) may be different than the techniques for correctingfor density. For example, various techniques for correcting a measuredMF_(apparent) to obtain an actual MF_(true) (or, at least,MF_(corrected)) are discussed in U.S. Pat. No. 6,505,519.

Examples of detailed techniques for correcting ρ_(apparent) andMF_(apparent) are discussed in more detail below. Generally speaking,though, with respect to FIG. 2, the digital transmitter is shown asincluding a density correction system 240, which has access to a densitycorrection database 245, and a mass flow rate correction system 250,which has access to a mass flow correction database 255. As discussed inmore detail below, the databases 245 and 255 may contain, for example,correction algorithms that have been derived theoretically or obtainedempirically, and/or correction tables that provide corrected density ormass flow values for a given set of input parameters. The databases 245and 255 also may store a variety of other types of information that maybe useful in performing the density or mass flow corrections. Forexample, the density correction database may store a number of densitiesρ_(liquid) corresponding to particular liquids (e.g., water or oil).

Further in FIG. 2, a void fraction determination/correction system 260is operable to determine a void fraction of a two-phase flow including aliquid and a gas. In one implementation, for example, the void fractiondetermination/correction system 260 may determine an actual voidfraction α_(true) from the corrected density ρ_(true). In anotherimplementation, the void fraction determination/correction system 260may input an apparent or indicated void fraction measurement obtained bythe void fraction sensor 235, and may correct this measurement based onan error characterization similar to the density and mass flowtechniques referred to above. In another implementation, the voidfraction sensor 235 may be operable to directly measure an actual voidfraction α_(true), in which case the void fractiondetermination/correction system 260 simply inputs this measurement.

Once the factors of ρ_(true), MF_(true), and α_(true) have beendetermined, and perhaps in conjunction with other known or discoverablequantities, a flow component mass flow rate determination system 265operates to simultaneously determine a mass flow rate for the liquidphase component and a mass flow rate for the gas phase component. Thatis, the transmitter 104 is operable to determine individual flowratesMF_(liquid) and MF_(gas) of the flow components, as opposed to merelydetermining the bulk flowrate of the combined or total two-phase flowMF_(true). Although, as just referred to, such measurements may bedetermined and/or output simultaneously, they also may be determinedseparately or independently of one another.

Once the component flow rates MF_(liquid) and MF_(gas) have beendetermined in the manner generally outlined above, these initialdeterminations may be improved upon by a process that relies onsuperficial velocities of the flow components, slip velocities betweenthe components, and/or an identified flow regime of the flow. In thisway, improved values for flow rates MF_(liquid) and MF_(gas) may beobtained, or may be obtained over time as those flow rates change.

Superficial velocities are referred to herein as those velocities thatwould exist if the same mass flow rate of a given phase was traveling asa single phase through the flowtube 215. A superficial velocitydetermination/correction system 270 is included in the transmitter 104for, for example, determining an apparent or corrected superficialvelocity of a gas or liquid in the two-phase flow.

Slip velocities refer to a condition in which gas and liquid phases in atwo-phase flow have different average velocities. That is, an averagevelocity of a gas AV_(gas) is different from an average velocity of aliquid AV_(liquid). As such, a phase slip S may be defined asS=AV_(gas)/AV_(liquid).

A flow regime is a term that refers to a characterization of the mannerin which the two phases flow through the flowtube 215 with respect toone another and/or the flowtube 215, and may be expressed, at leastpartially, in terms of the superficial velocities just determined. Forexample, one flow regime is known as the “bubble regime,” in which gasis entrained as bubbles within a liquid. As another example, the “slugregime” refers to a series of liquid “plugs” or “slugs” separated byrelatively large gas pockets. For example, in vertical flow, the gas ina slug flow regime may occupy almost an entire cross-sectional area ofthe flowtube 215, so that the resulting flow alternates betweenhigh-liquid and high-gas composition. Other flow regimes are known toexist and to have certain defined characteristics, including, forexample, the annular flow regime, the dispersed flow regime, and frothflow regime, and others.

The existence of a particular flow regime is known to be influenced by avariety of factors, including, for example, a gas void fraction in thefluid flow, an orientation of the flowtube 215 (e.g., vertical orhorizontal), a diameter of the flowtube 215, the materials includedwithin the two-phase flow, and the velocities (and relative velocities)of the materials within the two phase flow. Depending on these and otherfactors, a particular fluid flow may transition between several flowregimes over a given period of time.

Information about phase slip may be determined at least in part fromflow regime knowledge. For example, in the bubble flow regime, assumingthe bubbles are uniformly distributed, there may be little relativemotion between the phases. Where the bubbles congregate and combine toform a less uniform distribution of the gas phase, some slippage mayoccur between the phases, with the gas tending to cut through the liquidphase.

In FIG. 2, a flow regime determination system 275 is included that hasaccess to a database 280 of flow regime maps. In this way, informationabout an existing flow regime, including phase slip information, may beobtained, stored, and accessed for use in simultaneously determiningliquid and gas mass flow rates within a two-phase flow.

In FIG. 2, it should be understood that the various components of thedigital transmitter 104 are in communication with one another, althoughcommunication links are not explicitly illustrated, for the sake ofclarity. Further, it should be understood that conventional componentsof the digital transmitter 104 are not illustrated in FIG. 2, but areassumed to exist within, or be accessible to, the digital transmitter104. For example, the digital transmitter 104 will typically include(bulk) density and mass flow rate measurement systems, as well as drivecircuitry for driving the driver 210.

FIG. 3 is a flowchart 300 illustrating an operation of the coriolisflowmeter 200 of FIG. 2. Specifically, FIG. 3 illustrates techniques bywhich the flowmeter 200 of FIG. 2 is operable to simultaneouslydetermine liquid and gas flow rates MF_(liquid) and MF_(gas) for atwo-phase flow.

In FIG. 3, it is determined that a gas/liquid two-phase flow exists inthe flowtube 215 (302). This can be done, for example, by an operatorduring configuration of the mass flowmeter/densitometer for gas/liquidflow. As another example, this determination may be made automaticallyby using a feature of the coriolis meter to detect that a condition oftwo-phase gas-liquid flow exists. In the latter case, such techniquesare described in greater detail in, for example, U.S. Pat. No. 6,311,136and U.S. Pat. No. 6,505,519, incorporated by reference above.

Once the existence of two-phase flow is established, a corrected bulkdensity ρ_(true) is established (304) by the density correction system240, using the density correction database 245 of the transmitter 104.That is, an indicated density apparent is corrected to obtain ρ_(true).Techniques for performing this correction are discussed in more detailbelow.

Once ρ_(true) is determined, a corrected gas void fraction ρ_(true) maybe determined (306) by the void fraction determination/correction system260. Also, a corrected bulk mass flow rate MF_(true) is determined (308)by the mass flow rate correction system 250. As with density, techniquesfor obtaining the corrected void fraction α_(true) and mass flow rateMF_(true) are discussed in more detail below.

In FIG. 3, it should be understood from the flowchart 300 that thedeterminations of ρ_(true), α_(true), and MF_(true) may occur in anumber of sequences. For example, in one implementation, the correctedvoid fraction α_(true) is determined based on previously-calculatedcorrected density ρ_(true), whereupon the corrected mass flow rateMF_(true) is determined based on α_(true). In another implementation,α_(true) and ρ_(true) may be calculated independently of one another,and/or N_(ue) and MF_(true) may be calculated independently of oneanother.

Once corrected density ρ_(true), corrected void fraction α_(true), andcorrected mass flow rate MR_(true) are known, then the mass flow ratesof the gas and liquid components are determined (310) by the flowcomponent mass flow rate determination system 265. Techniques fordetermining the liquid/gas component flow rates are discussed in moredetail below with respect to FIG. 4.

Once determined, the liquid/gas component flow rates may be output ordisplayed (312) for use by an operator of the flowmeter. In this way,the operator is provided, perhaps simultaneously, with information aboutboth the liquid mass flow rate MF_(liquid) and the gas mass flow rateMF_(gas) of a two-phase flow.

In some instances, this determination may be sufficient (314), in whichcase the outputting of the liquid/gas component flow rates completes theprocess flow. However, in other implementations, the determination ofthe individual component mass flow rates may be improved upon byfactoring in information about, for example, the superficial velocitiesof the gas/liquid components, the flow regime(s) of the flow, and phaseslip, if any, between the components.

In particular, superficial velocities of the gas and liquid, SV_(gas)and SV_(liquid) are determined as follows. Gas superficial velocitySV_(gas) is defined as:

SV_(gas)=MF_(gas)/(ρ_(gas) *A _(T))  Eq. 1

where the quantity A_(T) represents a cross-section area of the flowtube215, which may be taken at a point where a void fraction of the flow ismeasured. Similarly, a liquid superficial velocity SV_(liquid) isdefined as:

SV_(liquid)=MF_(liquid)/(ρ_(liquid) *A _(T))  Eq. 2

As shown in Eqs. 1 and 2, determination of superficial velocities inthis context relies on the earlier determination of MF_(gas) andMF_(liquid). It should be understood from the above description and fromFIG. 3 that MF_(gas) and MF_(liquid) represent corrected or true massflow rates, MF_(gas) ^(true) and MF_(liquid) ^(true) since these factorsare calculated based on ρ_(true), α_(true), and MF_(true). As a result,the superficial velocities SV_(gas) and SV_(liquid) represent correctedvalues SV_(gas) ^(true) and SV_(liquid) ^(true). Further, the densityvalues ρ_(gas) and ρ_(liquid) refer, as above, to known densities of theliquid and gas in question, which may be stored in the densitycorrection database 245. As discussed in more detail below with respectto techniques for calculating corrected density ρ_(true), the densityvalues ρ_(gas) and ρ_(liquid) may be known as a function of existingtemperature or pressure, as detected by temperature sensor 220 andpressure sensor 225.

Using the superficial velocities and other known or calculated factors,some of which may be stored in the flow regime maps database 280, arelevant flow regime and/or phase slip may be determined (318) by theflow regime determination/correction system 275. Once superficialvelocities, flow regime, and phase slip are known, further correctionsmay be made to the corrected bulk density ρ_(true), corrected bulk massflow rate MF_(true), and/or corrected void fraction α_(true). In thisway, as illustrated in FIG. 3, component flow rates MF_(gas) andMF_(liquid) may be determined.

Flow regime(s) in two phase liquid/gas flow may be described by contourson a graph plotting the liquid superficial velocity versus the gassuperficial velocity. As just described, an improvement todeterminations of ρ_(true), α_(true), and/or MF_(true) may be obtainedby first establishing an approximate value of the liquid and gas flowrates, and then applying a more detailed model for the flow regimeidentified. For example, at relatively low GVF and relatively high flowthere exists a flow regime in which the aerated fluid behaves as ahomogenous fluid with little or no errors in both density and mass flow.This can be detected as homogenous flow requiring no correction, simplyusing observation of the drive gain, which shows little or no increasein such a setting, despite a significant drop in observed density.

FIG. 4 is a flowchart 400 illustrating techniques for determining liquidand gas flow rates MF_(liquid) and MF_(gas) for a two-phase flow. Thatis, the flowchart 400 generally represents one example of techniques fordetermining liquid and gas flow rates (310), as described above withrespect to FIG. 3.

In FIG. 4, the determination of liquid and gas flow rates (310) beginswith inputting the corrected density, void fraction, and mass flow ratefactors ρ_(true), α_(true), and MF_(true) (402). In a first instance,(404), the liquid and gas flow rates are determined (406) using Eqs. 3and 4:

MF_(gas)=α_(true)(ρ_(gas)/ρ_(true))(MF_(true))  Eq. 3

MF_(liquid)=(1−α_(true))(ρ_(liquid)/ρ_(true))(MF_(true))  Eq. 4

Eqs. 3 and 4 assume that there is no slip velocity (i.e., phase slip)between the liquid and gas phases (i.e., average velocity of the gasphase, AV_(gas), and average velocity of the liquid phase, AV_(liquid),are equal). This assumption is consistent with the fact that, in thefirst instance, superficial velocities and flow regimes (and therefore,phase slip) have not been determined.

In the second instance and thereafter (404), a determination is made,perhaps by the flow regime determination/correction system 275, as towhether phase slip exists (408). If not, then Eqs. 3 and 4 are usedagain (406) or the process ends.

If phase slip does exist (408), defined above as S=AV_(gas)/AV_(liquid),the terms MF_(gas) and MF_(liquid) are calculated using thecross-sectional area of the flowtube 215, A_(T), as also used in thecalculation of superficial velocities in Eqs. 1 and 2 (410). Using thedefinition of slip S just given,

MF_(gas)=ρ_(gas)(α_(true) A _(T))(AV_(gas))=ρ_(gas)(α_(true) A_(T))(S)(AV_(liquid))  Eq. 5

MF_(liquid)=ρ_(liquid)((1−α_(true))A _(T))(AV_(liquid))  Eq. 6

Since MF_(true)=MF_(gas)+MF_(liquid), Eqs. 5 and 6 may be solved forAV_(liquid) to obtain Eq. 7:

AV_(liquid)=MF_(true)/(A _(T)(ρ_(gas)α_(true) S+ρ_(liquid)(1−α_(true))))  Eq. 7

As a result, the liquid and gas flow rates are determined (406) usingEqs. 8 and 9:

MF_(liquid)=[ρ_(liquid)(1−α_(true))/(ρ_(gas)α_(true) S+ρ_(liquid)(1−α_(true)))][MF_(true)]  Eq. 8

MF_(gas)=MF_(true)−MF_(liquid)  Eq. 9

As described above, gas entrained in liquid forms a two-phase flow.Measurements of such a two-phase flow with a Coriolis flowmeter resultin indicated parameters ρ_(apparent), α_(apparent), and MF_(apparent)for density, void fraction, and mass flow rate, respectively, of thetwo-phase flow. Due to the nature of the two-phase flow in relation toan operation of the Coriolis flowmeter, these indicated values areincorrect by a predictable factor. As a result, the indicated parametersmay be corrected to obtain actual parameters ρ_(true), α_(true), andMF_(true). In turn, the actual, corrected values may be used tosimultaneously determine individual flow rates of the two (gas andliquid) components.

FIGS. 5A and 5B are graphs illustrating a percent error in a measurementof void fraction and liquid fraction, respectively. In FIG. 5A, thepercent error is a density percent error that is dependent on variousdesign and operational parameters, and generally refers to the deviationof the apparent (indicated) density from the true combined density thatwould be expected given the percentage (%) of gas in liquid.

In FIG. 5B, true liquid fraction versus indicated liquid fraction isillustrated. FIG. 5B shows the results, for the relevant flowmeterdesign, of several line sizes and flow rates. In more general terms, thefunctional relationship may be more complex and depend on both line sizeand flowrate. In FIG. 5B, a simple polynomial fit is shown that can beused to correct the apparent liquid fraction.

Other graphing techniques may be used; for example, true void fractionmay be plotted against indicated void fraction. For example, FIG. 6 is agraph illustrating a mass flow error as a function of a drop in densityfor a flowtube having a particular orientation and over a selected flowrange.

FIG. 7 is a flowchart 700 illustrating techniques for correcting densitymeasurements (304 in FIG. 3). In FIG. 7, the process begins with aninputting of the type of flowtube 215 being used (702), which mayinclude, for example, whether the flowtube 215 is bent or straight, aswell as other relevant facts such as a size or orientation of theflowtube 215.

Next, a gas-free density of the liquid, ρ_(liquid) is determined (704).This quantity may be useful in the following calculation(s), as well asin ensuring that that other factors that may influence the densitymeasurement ρ_(apparent), such as temperature, are not misinterpreted asvoid fraction effects. In one implementation, the user may enter theliquid density ρ_(liquid) directly, along with a temperature dependenceof the density. In another implementation, known fluids (and theirtemperature dependencies) may be stored in the density correctiondatabase 245, in which case the user may enter a fluid by name. In yetanother implementation, the flowmeter 200 may determine the liquiddensity during a time of single-phase, liquid flow, and store this valuefor future use.

An indicated mass flow rate MF_(apparent) is read from the Coriolismeter (706), and then an indicated density ρ_(apparent) is read from theCoriolis meter (708). Next, the density correction system 240 applieseither a theoretical, algorithmic (710) or empirical, tabular correction(712) to determine the true density ρ_(true) of the gas/liquid mixture.The quantity ρ_(true) may then be output as the corrected density (714).

An algorithmic density correction (710) may be determined based on theknowledge that, if there were no effect of the two-phase flow from thenormal operation of a Coriolis meter when used to measure density, theindicated density would drop by an amount derived from the equationdescribing void fraction, which is set forth above in terms of volumeflow and repeated here in terms of density as Eq. 10:

α_((%))=[(ρ_(apparent)−ρ_(liquid))/(ρ_(gas)−ρ_(liquid))]×100  Eq. 10

This can be used to define a quantity “density drop,” or Δρ, as shown inEq. 11:

Δρ=(ρ_(apparent)−ρ_(liquid))=α_((%))×(ρ_(gas)−ρ_(liquid))/100  Eq. 11

Note that Eq. 11 shows the quantity Δρ as being positive; however, thisquantity could be shown as a negative drop simply by multiplying theright-hand side of the equation by −1, resulting in Eq. 12:

Δρ=(ρ_(liquid)−ρ_(apparent))=α_((%))×(ρ_(liquid)−ρ_(gas))/100  Eq. 12

The quantity ρ_(gas) may be small compared to ρ_(liquid), in which caseEq. 12 may be simplified to Eq. 13:

Δρ=(ρ_(liquid)−ρ_(apparent))=α_((%))×ρ_(liquid)/100  Eq. 13

As discussed extensively above, density measurements by a Coriolismeter, or any vibrating densitometer, generally are under-reported bythe meter, and require correction. Accordingly, under two-phase flowEqs. 12 or 13 may thus be used to define the following two quantities: acorrected or true density drop, Δρ_(true), and an indicated or apparentdensity drop, Δρ_(app). Using Eq. 13 as one example, this results inEqs. 14 and 15:

Δρ_(true)=(ρ_(liquid)−ρ_(true))=α_((%))×ρ_(liquid)/100  Eq. 14

Δρ_(app)=(ρ_(liquid)−ρ_(apparent))=α_((%))×ρ_(liquid)/100  Eq. 15

There can be derived or empirically determined a relationship betweenΔρ_(true) and Δρ_(apparent) and apparent mass flow rate, MF_(apparent),as well as other parameters, such as, for example, drive gain, sensorbalance, temperature, phase regime, etc.). This relationship can beexpressed as shown as Δρ_(true)=f(MF_(apparent), drive gain, sensorbalance, temperature, phase regime, and/or other factors).

As a result, the relationship may generally be derived, or at leastproven, for each flowtube in each setting. For one model flowtube, knownand referred to herein as the Foxboro/Invensys CFS10 model flowtube, ithas been empirically determined that for some conditions the abovefunctional relationship can be simplified to be only a functionΔρ_(apparent) and of the form shown in Eq. 16:

$\begin{matrix}{{\Delta\rho}_{true} = {\sum\limits_{i = 0}^{M}{a_{i}\left( {\Delta\rho}_{apparent} \right)}^{i}}} & {{Eq}.\mspace{14mu} 16}\end{matrix}$

To force the condition for both sides of Eq. 16 to be zero when there isno apparent density drop relationship results in Eq. 17:

$\begin{matrix}{{\Delta\rho}_{true} = {\sum\limits_{i = 1}^{M}{a_{i}\left( {\Delta\rho}_{apparent} \right)}^{i}}} & {{Eq}.\mspace{14mu} 17}\end{matrix}$

M generally depends on the complexity of the empirical relationship, butin many cases can be as small as 2 (quadratic) or 3 (cubic).

Once the true density drop is determined, then working back through theabove equations it is straightforward to derive the true mixture densityρ_(true), as well as the true liquid and gas (void) fractions (thelatter being discussed in more detail with respect to FIG. 9).

A tabular correction for density (712) may be used when, for example, afunctional relationship is too complex or inconvenient to implement. Insuch cases, knowledge of the quantities Δρ_(apparent) and ΔMF_(apparent)may be used to determine Δρ_(true) by employing a table having the formof a table 800 of FIG. 8.

The table 800 may be, for example, a tabular look-up table that can be,for example, stored in the database 245, or in another memory, for useacross multiple applications of the table. Additionally, the table maybe populated during an initialization procedure, for storage in thedatabase 245 for an individual application of the table.

It should be understood that either or both of the algorithmic andtabular forms may be extended to include multiple dimensions, such as,for example, gain, temperature, balance, or flow regime. The algorithmicor tabular correction also may be extended to include other surfacefitting techniques, such as, for example, neural net, radical basisfunctions, wavelet analyses, or principle component analysis.

As a result, it should be understood that such extensions may beimplemented in the context of FIG. 3 during the approach describedtherein. For example, during a first instance, density may be determinedas described above. Then, during a second instance, when a flow regimehas been identified, the density may be further corrected using the flowregime information.

FIG. 9 is a flowchart 900 illustrating techniques for determining voidfraction measurements (306 in FIG. 3). In FIG. 9, the process beginswith an inputting by the void fraction determination system 240 of thepreviously-determined liquid and bulk (corrected) densities, ρ_(liquid)and ρ_(true) (902).

A density of the gas, ρ_(gas) is then determined (904). As with theliquid density ρ_(liquid), there are several techniques for determiningρ_(gas). For example, ρ_(gas) may simply be assumed to be a density ofair, generally at a known pressure, or may be an actual known density ofthe particular gas in question. As another example, this known densityρ_(gas) may be one of the above factors (i.e., known density of air orthe specific gas) at an actual or calculated pressure, as detected bythe pressure sensor 225, and/or at an actual or calculated temperature,as detected by the temperature sensor 220. The temperature and pressuremay be monitored using external equipment, as shown in FIG. 2, includingthe temperature sensor 220 and/or the pressure sensor 225.

Further, the gas may be known to have specific characteristics withrespect to factors including pressure, temperature, or compressibility.These characteristics may be entered along with an identification of thegas, and used in determining the current gas density ρ_(gas). As withthe liquid(s), multiple gasses may be stored in memory, perhaps alongwith the characteristics just described, so that a user may accessdensity characteristics of a particular gas simply by selecting the gasbe name from a list.

Once the factors ρ_(liquid), ρ_(gas), and ρ_(true) are known, then itshould be clear from Eq. 10 that void fraction α_(true) may be easilydetermined (906). Then, if needed, liquid fraction may be determined(908) simply by calculating 1−α_(true).

Although the above discussion presents techniques for determining voidfraction α_(true) based on density, it should be understood that voidfraction may be determined by other techniques. For example, anindicated void fraction α_(apparent) may be directly determined by theCoriolis flowmeter, perhaps in conjunction with other void fractiondetermination systems (represented by the void fraction sensor 235 ofFIG. 2), and then corrected based on empirical or derived equations toobtain α_(true). In other implementations, such external void fractiondetermining systems may be used to provide a direct measurement ofα_(true).

FIG. 10 is a flowchart 1000 illustrating techniques for determiningcorrected mass flow rate measurements (308 in FIG. 3). In FIG. 10, themass flow rate correction system 250 first inputs thepreviously-calculated corrected density drop Δρ_(true) (1002), and theninputs a measured, apparent mass flow rate MF_(apparent) (1004).

The mass flow rate correction system 250 applies either a tabular (1006)or algorithmic correction (1008) to determine the true mass flow rateMF_(true) of the gas/liquid mixture. The quantity MF_(true) may then beoutput as the corrected mass flow rate (1010).

In applying the tabular correction for mass flow rate (1006), knowledgeof the quantities Δρ_(true) and ΔMF_(apparent) may be used to determineMF_(true) by employing a table having the form of a table 1100 of FIG.11.

The table 1100, as with the table 800 may be, for example, a tabularlook-up table that can be, for example, stored in the database 245, orin another memory, for use across multiple applications of the table.Additionally, the table may be populated during an initializationprocedure, for storage in the database 255 for an individual applicationof the table.

Normalized values MF_(norm) _(—) _(app) and MF_(norm) _(—) _(true) maybe used in place of the actual ones shown above, in order to cover morethan one size coriolis flowtube. Also, the entries can be in terms ofthe correction, where the correction is defined by Eq. 18:

ΔMF=MF_(true)−MF_(apparent)  Eq. 18

The values in Eq. 18 should be understood to represent either actual ornormalized values.

In an algorithmic approach, as with density, the correction for massflow may be implemented by way of a theoretical or an empiricalfunctional relationship that is generally understood to be of the formΔMF=f(MF_(apparent), void fraction, drive gain, sensor balance,temperature, phase regime, and/or other factors).

For some cases the function can simplify to a polynomial, such as, forexample, the polynomial shown in Eq. 19:

$\begin{matrix}{{\Delta \; {MF}} = {\sum\limits_{i = 0}^{M}{\sum\limits_{j = 0}^{N}{a_{i}{b_{j}\left( {\Delta\rho}_{true}^{i} \right)}\left( {MF}_{norm\_ app}^{j} \right)}}}} & {{Eq}.\mspace{14mu} 19}\end{matrix}$

For some set of conditions, the functional relationship can be acombination of a polynomial and exponential, as shown in Eq. 20:

ΔMF=a ₁ de ^((a) ² ^(d) ² ^(+a) ³ ^(d+a) ⁴ ^(m) ² ^(+a) ⁵ ^(m)) +a ₆ d ²+a ₇ d+a ₈ m ² +a ₉ m  Eq. 20

In Eq. 20, d=Δρ_(true), and m=f(MF_(apparent)).

In one implementation, m in Eq. 20 may be replaced by apparentsuperficial liquid velocity SV_(liquid) which is given as describedabove by Eq. 2 as SV_(liquid)=MF_(liquid)(ρ_(liquid)*A_(T)). In thiscase, ρ_(liquid) and flowtube cross-section A_(T) are known or enteredparameters, and may be real-time corrected for temperature using, forexample, the on-board temperature measurement device 220 of the digitalcontroller/transmitter 104.

It should be understood that, as with the density corrections discussedabove, either or both of the algorithmic and tabular forms may beextended to include multiple dimensions, such as, for example, gain,temperature, balance, or flow regime. The algorithmic or tabularcorrection also may be extended to include other surface fittingtechniques, such as, for example, neural net, radical basis functions,wavelet analyses, or principle component analysis.

As a result, it should be understood that such extensions may beimplemented in the context of FIG. 3 during the approach describedtherein. For example, during a first instance, mass flow rate may bedetermined as described above. Then, during a second instance, when aflow regime has been identified, the mass flow rate may be furthercorrected using the flow regime information.

All of the above functional relationships for mass flow rate may berestated using gas fraction (α) or liquid fraction (100−α) instead ofdensity drop, as reflected in the table 1100 of FIG. 11. Also, althoughthe above described methods are dependent on knowledge of the correcteddensity drop Δρ_(true), it should be understood that other techniquesmay be used to correct an indicated mass flow rate. For example, varioustechniques for correcting mass flow rate measurements of a two-phaseflow are discussed in U.S. Pat. No. 6,505,519, incorporated by referenceabove.

Having described density, void fraction, and mass flow rate correctionsabove in general terms, for the purpose of, for example, simultaneouslycalculating individual flow component (phases) flow rates in a two-phaseflow, the below discussion and corresponding figures provide specificexamples of implementations of these techniques.

FIGS. 12-14 are graphs illustrating examples of density corrections fora number of flowtubes. In particular, the examples are based on dataobtained from three vertical water flowtubes, the flowtubes being: ½″,¾″, and 1″ in diameter.

More specifically, the ½″ data was taken with a 0.15 kg/s flow rate anda 0.30 kg/s flow rate; the ¾″ data was taken with a 0.50 kg/s flow rateand a 1.00 kg/s flow rate; and the 1″ data was taken with a 0.50 kg/sflow rate, a 0.90 kg/s flow rate, and a 1.20 kg/s flow rate. FIG. 12illustrates an error, e_(d), of the apparent density of the fluid-gasmixture (two-phase flow) versus the true drop in density of thefluid-gas mixture, Δρ_(true):

$\begin{matrix}{{\Delta\rho}_{true} = {100 \cdot \frac{\rho_{liquid} - \rho_{true}}{\rho_{liquid}}}} & {{Eq}.\mspace{14mu} 21} \\{e_{d} = {100 \cdot \frac{\rho_{apparent} - \rho_{true}}{\rho_{true}}}} & {{Eq}.\mspace{14mu} 22}\end{matrix}$

where, as above, ρ_(liquid) is the density of the gas-free liquid,ρ_(true) is the true density of the liquid-gas mixture, and ρ_(apparent)is the apparent or indicated density of the liquid-gas mixture.

In FIG. 12, the correction is performed in terms of the apparent drop inmixture density, Δρ_(apparent), as shown in Eq. 23:

$\begin{matrix}{{\Delta\rho}_{apparent} = {100 \cdot \frac{\rho_{liquid} - \rho_{apparent}}{\rho_{apparent}}}} & {{Eq}.\mspace{14mu} 23}\end{matrix}$

In FIG. 12, when fitting the data, both the apparent and true drop indensity of the mixture were normalized to values between 0 and 1 bydividing them through by 100, where this normalization is designed toensure numerical stability of the optimization algorithm. In otherwords, the normalized apparent and true drop in mixture density are theapparent and true drop in mixture density defined as a ratio, ratherthan as a percentage, of the liquid density ρ_(liquid), as shown in Eq.24:

$\begin{matrix}{{\Delta\rho}_{apparent}^{normalized} = \frac{{\Delta\rho}_{apparent}}{100}} & {{Eq}.\mspace{14mu} 24}\end{matrix}$

The model formula, based on Eq. 17, provides Eq. 25:

Δρ_(true) ^(normalized) =a ₁(Δρ_(apparent) ^(normalized))³ +a₂(Δρ_(apparent) ^(normalized))² +a ₃(Δρ_(apparent) ^(normalized))  Eq.25

In this case, the coefficients are a₁=−0.51097664273685,a₂=−1.26939674868129, and a₃=0.24072693119420. FIGS. 13A and 13Billustrate the model with the experimental data and the residual errors,as shown. FIGS. 14A and 14B give the same information, but with eachflow rate plotted separately.

To summarize, the drop in density correction is performed in thetransmitter 104 by calculating the apparent density drop Δρ_(apparent),using the apparent density value ρ_(apparent) and the liquid densityρ_(liquid). The value of the apparent drop in density is normalized toobtain

${{\Delta\rho}_{apparent}^{normalized} = \frac{{\Delta\rho}_{apparent}}{100}},$

so that, as explained above, the drop in density is calculated as aratio rather than a percentage. The density correction model(s) may thenbe applied to obtain the normalized corrected drop in mixture densityΔπ_(true) ^(normalized). Finally, this value is un-normalized to obtainthe corrected drop in density Δρ_(true)=100·Δρ_(true) ^(normalized). Ofcourse, the final calculation is not necessary if the corrected drop inmixture density Δρ_(true) is defined as a ratio rather than percentageof the true value.

FIGS. 15-20 are graphs illustrating examples of mass flow ratecorrections for a number of flowtubes. In particular, the examples arebased on data obtained from three vertical water flowtubes, theflowtubes being: ½″, ¾″, and 1″ in diameter. More specifically, the ½″data was taken with a 0.15 kg/s flow rate and a 0.30 kg/s flow rate; the¾″ data was taken with a 0.50 kg/s flow rate and a 1.00 kg/s flow rate;and the 1″ data was taken with 18 flow rates between 0.30 kg/s and 3.0kg/s flow rate, with a maximum drop in density of approximately 30%.

FIGS. 15A and 15B illustrate apparent mass flow errors for the data usedto fit the model versus corrected drop in mixture density Δρ_(true) andnormalized true superficial fluid velocity; i.e., the apparent mass flowerror curves per flowline, together with a scatter plot of the apparentmass flow error versus corrected drop in density Δρ_(true) andnormalized true superficial fluid velocity ν_(tn), as shown in Eq. 26:

$\begin{matrix}{{v_{tn} = \frac{v_{t}}{v_{\max}}},\mspace{14mu} {v_{t} = \frac{m_{t}}{\rho_{liquid} \cdot A_{T}}}} & {{Eq}.\mspace{14mu} 26}\end{matrix}$

where m_(t) is the true fluid mass flow, i.e. the value of the mass flowindependently measured, ρ_(liquid) is the liquid density, A_(T) is theflowtube cross-section area, and ν_(max) is the maximum value for thesuperficial fluid velocity (here considered 12), so that ν_(tn) givesthe ratio of the true superficial fluid velocity from the whole range ofthe flowtube 215. In these examples, both drop in mixture density andsuperficial fluid velocity are normalized between 0 and 1 prior tofitting the model, for the purpose of ensuring numerical stability forthe model optimization algorithm.

FIG. 16 illustrates apparent mass flow errors versus corrected drop inmixture density and normalized apparent superficial fluid velocity, withsafety bounds for the correction mode. That is, FIG. 16 gives thescatter plot of the apparent mass flow errors versus corrected drop indensity and, this time, normalized apparent superficial fluid velocity

${v_{n} = {\frac{v}{v_{\max}} = \frac{m}{v_{\max} \cdot \rho \cdot A}}},$

where m is the apparent fluid mass flow (i.e. as measured by thetransmitter 104). Superimposed on the plot are the boundaries definingthe safe region for the model, i.e., the region for which the model isexpected to give an accuracy similar with the one for the fit data.Using this nomenclature, the apparent mass flow error e is given by

$e = {100 \cdot {\frac{m - m_{t}}{m_{t}}.}}$

The model formula for this situation is shown as Eq. 27:

e _(n) =a ₁ dd _(cn) ·e ^(a) ² ^(dd) ^(cn) ² ^(+a) ³ ^(dd) ^(cn) ^(+a) ⁴^(ν) ^(n) ² ^(+a) ⁵ ^(ν) ^(n) +a ₆ dd _(cn) ² +a ₇ dd _(cn) +a ₈ν_(n) ²a ₉ν_(n)  Eq. 27

where

$\begin{matrix}{e_{n} = {\frac{e}{100} = \frac{m - m_{t}}{m_{t}}}} & {{Eq}.\mspace{14mu} 28}\end{matrix}$

where, in Eqs. 27 and 28, dd_(cn) is the normalized corrected drop inmixture density, and ν_(n) is the normalized apparent superficialvelocity of the liquid.

In this case, the coefficient are: a₁=−4.78998578570465,a₂=4.20395000016874, a₃=−5.93683498873342, a₄=12.03484566235777,a₅=−7.70049487145105, a₆=0.69537907794202, a₇=−0.52153213037389,a₈=0.36423791515369, and a₉=−0.16674339233364

FIG. 17 illustrates a scatter plot for the model residuals, togetherwith the model formula and coefficients; i.e., shows model residualsversus the corrected drop in mixture density and normalized true fluidvelocity. FIGS. 18A-18D and FIGS. 19A-19D give the model residual errorsfor the whole data set used to fit the model and the actual data alone,respectively. Finally, FIGS. 20A and 20B illustrate the model surfaceboth interpolating and extrapolating outside the safe fit area. FromFIGS. 16, 20A, and 20B, the apparent mass flow (superficial liquidvelocity) and drop in density bounds for the model should be understood.

To summarize, mass flow correction in the transmitter 104 is undertakenin this example by calculating an apparent drop in density, correctingit using the method(s) described above, and normalizing the resultingvalue by dividing it by 100 (or use the obtained normalized correcteddrop in density from the density model). Then, a normalized superficialfluid velocity ν_(n) is calculated, and the model is applied to obtainan estimation of the normalized mass flow error e_(n), where this valuegives the error of the apparent mass flow as a ratio of the true massflow. The obtained value may be un-normalized by multiplying it by 100,to thereby obtain the mass flow error as a percentage of the true massflow. Finally, the apparent mass flow may be corrected with theun-normalized mass flow error

$m_{c} = {\frac{m}{e_{n} + 1}.}$

As will be appreciated, the above description has a wide range ofapplications to improve the measurement and correction accuracy of acoriolis meter during two phase flow conditions. In particular, thetechniques described above are particularly useful in measurementapplications where the mass flow of the liquid phase and the mass flowof the gas phase must be measured and/or corrected to a high level ofaccuracy. One exemplary application is the measurement of the mass flowof the liquid phase and the measurement of the gas phase in oil and gasproduction environments.

The above discussion is provided in the context of the digital flowmeterof FIG. 2. However, it should be understood that any vibrating oroscillating densitometer or flowmeter, analog or digital, that iscapable of measuring multi-phase flow that includes a gas phase of acertain percentage may be used. That is, some flowmeters are onlycapable of measuring process fluids that include a gas phase when thatgas phase is limited to a small percentage of the overall process fluid,such as, for example, less than 5%. Other flowmeters, such as thedigital flowmeter(s) referenced above, are capable of operation evenwhen the gas void fraction reaches 40% or more.

Many of the above-given equations and calculations are described interms of density, mass flow rate, and/or void fraction. However, itshould be understood that the same or similar results may be reachedusing variations of these parameters. For example, instead of mass flow,a volumetric flow may be used. Additionally, instead of void fraction,liquid fraction may be used.

A number of implementations have been described. Nevertheless, it willbe understood that various modifications may be made. Accordingly, otherimplementations are within the scope of the following claims.

1. A flowmeter comprising: a vibratable flowtube; a driver connected tothe flowtube and operable to impart motion to the flowtube; a sensorconnected to the flowtube and operable to sense the motion of theflowtube and generate a sensor signal; and a controller connected toreceive the sensor signal, the controller being operable to determine afirst flow rate of a first phase within a two-phase flow through theflowtube and determine a second flow rate of a second phase within thetwo-phase flow. 2-20. (canceled)